# Standard Form Calculator

By Maciej Kowalski, PhD candidate
Last updated: Apr 08, 2021

Welcome to the standard form calculator, where we'll learn how to write a number in standard form. "What is the standard form?" Well, we'll get to the standard form definition soon enough. But let's just say that standard form in math and physics (quite often called scientific notation) is a neat way of dealing with very large or very small values. It's quite troublesome to write all the zeros of a number in every line of our calculations. Preferably, we can use standard form exponents and write the same thing with just a few symbols. That's why we made this standard form converter - to help you with just that.

For our non-American friends out there, the standard form is usually a quite different thing. Outside of the USA (especially in the UK), we say that a number is in its standard form if it's a single value that involves no arithmetic operations whatsoever. This notion is connected to the expanded form, and we explain it all in detail in the dedicated section. Also, note how you can switch between the two variants in the advanced section by choosing the appropriate option in the field "Have the calculator use..."

Warning: There is a possibility that you've come across this standard form calculator in search of different ways of writing a quadratic equation. In that case, check out our quadratic formula calculator, but be sure to come back to us whenever you get the chance!

Now, back to the question of the hour, "What does standard form mean?"

## What is the standard form?

As we've already said above, writing numbers in standard form is useful whenever we have some values that are either very large, or very small. Formally, the standard form definition is

`a = b * 10ⁿ`,

where,

• `a` is the number we want to convert to standard form;
• `b` is a value between `1` and `10` (including `1` but excluding `10`); and
• `n` is some integer (possibly negative).

So what does standard form mean? In essence, it tells you to take the significant figures that tell you (with some precision) what the value is, and then separately write the order of magnitude using the standard form exponents. To give you an example of how we use the standard form definition in real life, recall the stimulus check given to American citizens by the authorities and how much money it involved. It's much easier to write (and read) that it was a \$2 trillion, rather than a \$2,000,000,000,000 package, wouldn't you say?

Alright, let's now take a look at how we convert to standard form.

## What does standard form mean?

Before we give some examples of writing numbers in standard form in physics or chemistry, let's recall from the above section the standard form math formula:

`a = b * 10ⁿ`.

We said that the number `b` should be between `1` and `10`. This means that, for example, `1.36 * 10⁷` or `9.81 * 10⁻²³` are in standard form, but `13.1 * 10¹²` isn't because `13.1` is bigger than `10`. We could, however, convert it to standard form by saying that

`13.1 * 10¹² = 1.31 * 10¹³`.

What we've done above is basically move the point, `.`, that separates the digits of the number by one place, which is equivalent to `10` to the power of one. And that is exactly what the standard form exponents are for.

If we take a number, say, `12,345.6789`, and multiply it by `10`, we'll get

`12,345.6789 * 10¹ = 123,456.789`.

In other words, we move the point one place to the right. If we do it again, we'll get

`(12,345.6789 * 10¹) * 10¹ = 12,345.6789 * 10² = 1,234,567.89`,

which is the number we had initially but with the point two places to the right. This movement by `2` is shown by the power in the standard form exponents.

Conversely, if we divide the initial number by `10`, which is equal to multiplying it by `1/10 = 10⁻¹`, we'll get

`12,345.6789 * 10⁻¹ = 1,234.56789`,

which is the value we had but with the point one place to the left.

To sum up, in the standard form math formula

`a = b * 10ⁿ`,

the absolute value of the standard form exponents `n` tell us how many places we have to move the point, and the sign of `n` indicates if it should be to the right (for `n` positive) or the left (for `n` negative). Therefore, when we convert to standard form, it's all about choosing the power of `10` in such a way so that the `b` in the formula is between `1` and `10`.

Alright, after all this time learning the theory, it's finally time we saw some standard form examples, wouldn't you say?

## Standard form in connection with expanded form

Non-Americans often refer to the standard form in math in connection with a very different topic. To be precise, they understand it as the basic way of writing numbers (with decimals) using the decimal base (as opposed to, say, the binary base), which we can decompose into terms representing the consecutive digits.

For instance, take the number `154.37`. It is in its standard form in the decimal base. That means `1` is the hundreds digit, `5` is that of tens, `4` of ones, `3` of tenths, and `7` of hundredths. Having the number written the way it is, makes us see it as a whole, and we don't really think of the individual digits, do we?

The expanded form is a way to write a number as a sum, each summand corresponding to one of the number's digits. In our case, the sum would be:

`154.37 = 100 + 50 + 4 + 0.3 + 0.07`.

As you can see, we had five digits, so we get five terms. What is more, consecutive digits appear in consecutive summands; we simply add a few zeros in the correct places to make it all jump to the right spot when we add it all up.

Still, we might wish to decompose it even further. After all, we wanted to see the digits themselves (i.e., as one-digit numbers), and not some "complicated" expression like `0.07`. Therefore, we can also write:

`154.37 = 1*100 + 5*10 + 4*1 + 3*0.1 + 7*0.01`.

It might seem artificial to write sum of the products, like `1*100` or `4*1`, but that's just what the expanded form is.

This time, we indeed see the digits as the first factors in each multiplication. Moreover, the second factors have a lot in common - they consist of a single `1` with some zeros (possibly none).

The sum we got can encourage us to go even further! After all, we can get `100`, `10`, `1`, `0.1`, and `0.01` by raising the number `10` to integer powers: to the power `2`, `1`, `0`, `-1`, and `-2`, respectively. In other words, we can also write:

`154.37 = 1*10² + 5*10¹ + 4*10⁰ + 3*10⁻¹ + 7*10⁻²`.

The three decompositions we got in this section are all expanded forms: using numbers, factors, and exponents. Our standard form calculator allows you to have any of them - just pick the variant you like best under "Show the number in ... form."

## Standard form examples

Now that we've seen how to write a number in standard form, it's time to convince you that it's a useful thing to do. Of course, we know that you're most probably learning all of this for the pure pleasure of grasping yet another part of theoretical mathematics, but it doesn't hurt to take a look at physics or chemistry from time to time. You know, those two minor branches of mathematics.

In the first section, we mentioned that the standard form converter is most useful when we're dealing with very large or very small numbers. So, why don't we take one object from each side of the spectrum: a planet and an atom.

The mass of the Earth is, approximately,

`5,972,000,000,000,000,000,000,000 kg`

and its circumference is

`40,075 km`.

Don't ask us how they found the mass of the Earth, there aren't any scale big enough to weigh the entire planet. As for the circumference, talk to Eratosthenes.

Anyway, if scientists had to write all of those zeros every time they calculated something about our planet, they'd waste ages! It's much easier to recall how to write a number in standard form and say that the mass of Earth is, in fact,

`5.972 * 10²⁴ kg`

and the circumference is... actually, the `40,075 km` doesn't look that bad, does it? Well, we could change it into `4.0075 * 10⁴ km`, but is it better that way? If we needed to change it to millimeters, then maybe it'd be a better idea, but the kilometer form seems perfectly usable.

There is a valuable lesson here: writing numbers in standard form is not always the way to go. It's all about simplicity of notation, but, at the end of the day, it pretty much boils down to a matter of personal preference (or your teacher's if you're writing a test).

Back to the standard form examples, the mass of a helium atom is (approximately)

`0.0000000000000000000000000066423 kg`

`0.00000000014 m`.

Now, this looks even worse than the previous example, it doesn't have the commas in between! Thankfully, there are tools - like our standard form calculator - to make our lives easier. So, what is the standard form of the above numbers?

`6.6423 * 10⁻²⁷ kg` and

`1.4 * 10⁻¹⁰ m`.

Arguably, there is now no question that writing the numbers in standard form was a good idea.

## Example: using the standard form calculator

Suppose that you've taken up astronomy recently, and would like to know the gravitational force acting between the Earth and the Moon. For the calculations, we need the masses of the two objects (denote the Earth's by `M₁` and the Moon's by `M₂`) and the distance between them (denoted by `R`). We have

`M₁ = 5,972,000,000,000,000,000,000,000 kg`,

`M₂ = 73,480,000,000,000,000,000,000 kg`, and

`R = 384,400,000 m`.

If you look at our gravitational force calculator, you'll find that we'll be using the formula

`F = G * M₁ * M₂ / R²`,

where

• `G` is the gravitational constant, `G = 0.00000000006674 N*m²/kg²`.

All in all, the gravitational force is

`F = 0.00000000006674 N*m²/kg² * 5,972,000,000,000,000,000,000,000 kg * 73,480,000,000,000,000,000,000 kg / (384,400,000 m)²`.

Um... maybe we should reconsider that and try our standard form calculator. What do you say?

Let's input the four above numbers into our standard form converter. It will tell us that

`M₁ = 5.972 * 10²⁴ kg`,

`M₂ = 7.348 * 10²² kg`,

`R = 3.844 * 10⁸ m`, and

`G = 6.674 * 10⁻¹¹ N*m²/kg²`.

All in all, we can rewrite our formula as

`F = 6.674 * 10⁻¹¹ N*m²/kg² * 5.972 * 10²⁴ kg * 7.348 * 10²² kg / (3.844 * 10⁸ m)²`.

Now, this is more like it! We don't know about you, but for us short is beautiful, in mathematics at least.

But there's more! We have multiplication and division in the formula, and the standard form exponents make these two operations very easy to calculate. By the well-known, well-remembered, and totally not forgotten the moment the test was over formulas, multiplying two powers with the same base is the same as adding the exponents, while dividing corresponds to subtracting them. In other words, if we separate the `10`'s to some powers from the other numbers, we'll get

`F = (6.674 * 5.972 * 7.348 / 3.844²) * (10⁻¹¹ * 10²⁴ * 10²² / (10⁸)²) N`,

which is

`F = (6.674 * 5.972 * 7.348 / 3.844²) * 10⁻¹¹⁺²⁴⁺²²⁻¹⁶ N`.

Now it's just a few good old calculations to get:

`F ≈ 19.82 * 10¹⁹ N`.

We've spent quite some time together with the standard form calculator, enough to know that we can't leave the answer like this. We haven't learned how to write a number in standard form for nothing.

`F ≈ 1.982 * 10²⁰ N`.

Now that we know with what force the Earth attracts the Moon, maybe we could do something similar to get the attraction between you and your crush?

Maciej Kowalski, PhD candidate
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